09757 - Geometry and Algebra

Learning outcomes

At the end of the course, the student has all the essential knowledge about linear algebra. More precisely, the student will be able to solve systems of linear equations as well as work with vector spaces and linear maps (for example they will be able to compute eigenvalues, eigenvectors and eigenspaces of linear endomorphisms).

Course contents

The aim of the course is to provide the essential tools for the study of systems of linear equations and the linear algebra.

The course will cover the following topics:

1) Introduction to systems of linear equations:

• Definitions and examples;
• Matrices associated to a given system of linear equations;
• Gaussian elimination;
• Solutions of systems of linear equations depending on a given real parameter.

2) Introduction to vector spaces:

• Definitions and examples;
• The vector space Rn and its geometric interpretation;
• Additional examples of vector spaces (defined in terms of matrices and polynomials).

3) Introduction to vector subspaces:

• Definitions, examples and counterexamples;
• Linear combination of vectors;
• Generators of a vector space;
• Linear independence;
• Basis and dimension of a vector space;
• Some criteria for the linear independence.

4) Introduction to sums and intersections of vector spaces:

• Definitions of sums, unions and intersections of vector spaces;
• Grassmann formula;
• Direct sum;
• Examples.
  

5) Coordinates of a vector with respect to a given basis:

• Definition;
• Change of basis;
• Examples.
  

6) Introduction to linear maps:

• Definitions and examples;
• Existence and uniqueness;
• Kernel and image of a linear map;
• Injectivity and surjectivity of a linear map;
• Rank-Nullity Theorem;
• Matrix associated to a linear map;
• Rank of a matrix;
• Composition of linear maps (and matrices);
• Invertible linear maps and invertible matrices;
• How to express the same linear map in two distinct bases.

7) More on linear maps (and matrices):

• Matrix similarity;
• Determinant of a matrix;
• Eigenvalues and eigenvectors of a linear endomorphism;
• Eigenspaces;
• Characteristic polynomial;
• Diagnolizable matrix and diagonalizable linear endomorphism;
• Examples;
• Linear endomorphisms which depend on some given real parameter and detect when they are diagonalizable.

Readings/Bibliography

Books:

• Marco Abate "Algebra Lineare" (McGraw-Hill, 2000)
  

• Stefano Francaviglia "Geometri e Algebra T. (Aggiornamento 2017)" ISBN 9780244020736

Exercises:

• Stefano Francaviglia "Test di Algebra Lineare" ISBN 9781540411921

Teaching methods

The course is organized in 60 hours of in-person teaching. Each lecture will contain new aspects of the theory as well as many examples/exercises. This will help the students to become more familiar with the new definitions. Moreover, every week the students will get access to a collection of exercises in order to test their understanding.

 

The students will find the order of the topics covered in this course very convenient. Indeed, they will learn how to prove a statement/theorem beginning with some elementary cases and ending with the most relevant results in the course.

Assessment methods

The exam consists of a written exam and an oral exam.

The written exam will consist mainly of exercises plus a short question about the theory.

The candidate who has solved in a "sufficient" way the exercise part is admitted to the oral exam.

The length of the oral exam (in which the knowledge of the theory is evaluated) depends on the correctness of the answer to the question about the theory in the written exam (that means that the oral exam will be short if the question in the written exam has been answered in a correct way).

Teaching tools

In the "Virtuale" online platform, the students will find some lecture notes about the topics covered in the course. Moreover, they will also find weekly exercise sheets (together with their solutions that will be uploaded a week later than the corresponding exercises). Finally, they will also get access to the written exams of the previous years together with their solutions.

Students can arrange some online meetings with the teacher (scheduled in advance via email) in order to ask questions about the theory/exercises.

Office hours

See the website of Marco Moraschini